Estimating and Modelling the Proportion Cured of Disease in Population Based Cancer Studies

Transcription

1 Estimating and Modelling the Proportion Cured of Disease in Population Based Cancer Studies Paul C Lambert Centre for Biostatistics and Genetic Epidemiology, University of Leicester, UK 12th UK Stata Users Group Meeting London 11th-12th September 2006 Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 1/27

2 Population Based Cancer Studies Using data from cancer registries. Attempt to obtain all diagnosed cancers. Information used for incidence and survival. Large sample sizes. Relative Survival methods used for survival analysis. Five year relative survival often reported. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 2/27

3 Relative Survival Relative Survival = Observed Survival Expected Survival R(t) = S(t)/S (t) Expected survival obtained from national population life tables stratified by age, sex, year of diagnosis, other covariates. Estimate of mortality associated with a disease without requiring information on cause of death. On hazard scale h(t) = h (t) + λ(t) Observed Mortality Rate = Expected Mortality Rate + Excess Mortality Rate Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 3/27

4 Relative Survival Relative Survival = Observed Survival Expected Survival R(t) = S(t)/S (t) Expected survival obtained from national population life tables stratified by age, sex, year of diagnosis, other covariates. Estimate of mortality associated with a disease without requiring information on cause of death. On hazard scale h(t) = h (t) + λ(t) Observed Mortality Rate = Expected Mortality Rate + Excess Mortality Rate Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 3/27

5 Relative Survival Relative Survival = Observed Survival Expected Survival R(t) = S(t)/S (t) Expected survival obtained from national population life tables stratified by age, sex, year of diagnosis, other covariates. Estimate of mortality associated with a disease without requiring information on cause of death. On hazard scale h(t) = h (t) + λ(t) Observed Mortality Rate = Expected Mortality Rate + Excess Mortality Rate Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 3/27

6 Relative Survival Models Usually model on the log excess hazard (mortality) scale[3]. h(t) = h (t) + exp(βx) Parameters are (log) excess hazard ratios. Models have proportional excess hazards as a special case, but often non-proportional excess hazards are observed. Non-proportionality modelled piecewise[3], using fractional polynomials[6], or splines[4]. The models do not assume that a proportion of patients may be cured of their disease. For details of Stata command strs for estimation and modelling of relative survival using piecewise methods see Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 4/27

7 Relative Survival Models Usually model on the log excess hazard (mortality) scale[3]. h(t) = h (t) + exp(βx) Parameters are (log) excess hazard ratios. Models have proportional excess hazards as a special case, but often non-proportional excess hazards are observed. Non-proportionality modelled piecewise[3], using fractional polynomials[6], or splines[4]. The models do not assume that a proportion of patients may be cured of their disease. For details of Stata command strs for estimation and modelling of relative survival using piecewise methods see Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 4/27

8 Relative Survival Models Usually model on the log excess hazard (mortality) scale[3]. h(t) = h (t) + exp(βx) Parameters are (log) excess hazard ratios. Models have proportional excess hazards as a special case, but often non-proportional excess hazards are observed. Non-proportionality modelled piecewise[3], using fractional polynomials[6], or splines[4]. The models do not assume that a proportion of patients may be cured of their disease. For details of Stata command strs for estimation and modelling of relative survival using piecewise methods see Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 4/27

9 Relative Survival Models Usually model on the log excess hazard (mortality) scale[3]. h(t) = h (t) + exp(βx) Parameters are (log) excess hazard ratios. Models have proportional excess hazards as a special case, but often non-proportional excess hazards are observed. Non-proportionality modelled piecewise[3], using fractional polynomials[6], or splines[4]. The models do not assume that a proportion of patients may be cured of their disease. For details of Stata command strs for estimation and modelling of relative survival using piecewise methods see Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 4/27

10 Relative Survival Models Usually model on the log excess hazard (mortality) scale[3]. h(t) = h (t) + exp(βx) Parameters are (log) excess hazard ratios. Models have proportional excess hazards as a special case, but often non-proportional excess hazards are observed. Non-proportionality modelled piecewise[3], using fractional polynomials[6], or splines[4]. The models do not assume that a proportion of patients may be cured of their disease. For details of Stata command strs for estimation and modelling of relative survival using piecewise methods see Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 4/27

11 Definition of Cure (1) For many cancers the hazard (mortality) rate returns to the same level as that in the general population. When this occurs the relative survival curve is seen to reach a plateau (or the excess hazard rate approaches zero). This is Population or Statististical Cure. Information of cure at the individual level not available. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 5/27

12 Definition of Cure (1) For many cancers the hazard (mortality) rate returns to the same level as that in the general population. When this occurs the relative survival curve is seen to reach a plateau (or the excess hazard rate approaches zero). This is Population or Statististical Cure. Information of cure at the individual level not available. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 5/27

13 Definition of Cure (1) For many cancers the hazard (mortality) rate returns to the same level as that in the general population. When this occurs the relative survival curve is seen to reach a plateau (or the excess hazard rate approaches zero). This is Population or Statististical Cure. Information of cure at the individual level not available. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 5/27

14 Definition of Cure (1) For many cancers the hazard (mortality) rate returns to the same level as that in the general population. When this occurs the relative survival curve is seen to reach a plateau (or the excess hazard rate approaches zero). This is Population or Statististical Cure. Information of cure at the individual level not available. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 5/27

15 Definition of Cure (2) Relative Survival Time from Diagnosis (Years) Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 6/27

16 Definition of Cure (2) Relative Survival Time from Diagnosis (Years) Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 6/27

17 Relative Survival for Cancer of the Colon in Finland Relative Survival Years from Diagnosis Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 7/27

18 Mixture and Non-Mixture Models Relative Survival Models S(t) = S (t)r(t) h(t) = h (t) + λ(t) When modelling cure we define an asymptote at the cure fraction, π, for the relative survival function, R(t). The excess hazard rate, λ(t), has an asymptote at zero. Two main approaches Mixture Model Non-Mixture Model Both of these models have been used in standard survival analysis [9], i.e. not incorporating background mortality. Some of these models are implemented in Stata using the cureregr command. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 8/27

19 Mixture and Non-Mixture Models Relative Survival Models S(t) = S (t)r(t) h(t) = h (t) + λ(t) When modelling cure we define an asymptote at the cure fraction, π, for the relative survival function, R(t). The excess hazard rate, λ(t), has an asymptote at zero. Two main approaches Mixture Model Non-Mixture Model Both of these models have been used in standard survival analysis [9], i.e. not incorporating background mortality. Some of these models are implemented in Stata using the cureregr command. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 8/27

20 Mixture Model Mixture Model S(t) = S (t)(π + (1 π)s u (t)) h(t) = h (t) + (1 π)fu(t) π+(1 π)s u(t) S (t) is the expected survival. π is the proportion cured (the cure fraction). (1 π) is the proportion uncured (those bound to die ). S u (t) is the survival for the uncured group. See De Angelis [2] and Verdecchia [10] for more details. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 9/27

21 Non-Mixture Model Non Mixture Model S(t) = S (t)π Fz(t) h(t) = h (t) ln(π)f z (t) We have extended the non-mixture model to relative survival[7]. If parameters in f z (t) do not vary by covariates then this is a proportional excess hazards model. The mixture model does not have proportional excess hazards as a special case. The non-mixture model can also be written as; S(t) = S (t) ( π + (1 π) ( π Fz(t) π) 1 π )) This is a mixture cure fraction model and thus the survival function of uncured patients can also be obtained from a non-mixture model by a simple transformation of the model parameters. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 10/27

22 Non-Mixture Model Non Mixture Model S(t) = S (t)π Fz(t) h(t) = h (t) ln(π)f z (t) We have extended the non-mixture model to relative survival[7]. If parameters in f z (t) do not vary by covariates then this is a proportional excess hazards model. The mixture model does not have proportional excess hazards as a special case. The non-mixture model can also be written as; S(t) = S (t) ( π + (1 π) ( π Fz(t) π) 1 π )) This is a mixture cure fraction model and thus the survival function of uncured patients can also be obtained from a non-mixture model by a simple transformation of the model parameters. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 10/27

23 Likelihood Relative Survival Models L i = d i ln(h (t i )+λ(t i ))+ ln(s (t i ))+ln(r(t i )) ln(s (t 0i )) ln(r(t 0i )) S (t i ) and S (t 0i ) do not depend on the model parameters and can be excluded from the likelihood. Merge in expected mortality rate at time of death, h (t i ). Newton-Raphson algorithm implemented using Stata ml command (method lf). Incorporating delayed entry allows period analysis models to be fitted[8]. This is a method used to obtain up-to-date estimates of (relative) survival. Application in the cure models allows up-to-date estimates of cure to be obtained. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 11/27

24 Likelihood Relative Survival Models L i = d i ln(h (t i )+λ(t i ))+ ln(s (t i ))+ln(r(t i )) ln(s (t 0i )) ln(r(t 0i )) S (t i ) and S (t 0i ) do not depend on the model parameters and can be excluded from the likelihood. Merge in expected mortality rate at time of death, h (t i ). Newton-Raphson algorithm implemented using Stata ml command (method lf). Incorporating delayed entry allows period analysis models to be fitted[8]. This is a method used to obtain up-to-date estimates of (relative) survival. Application in the cure models allows up-to-date estimates of cure to be obtained. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 11/27

25 Likelihood Relative Survival Models L i = d i ln(h (t i )+λ(t i ))+ ln(s (t i ))+ln(r(t i )) ln(s (t 0i )) ln(r(t 0i )) S (t i ) and S (t 0i ) do not depend on the model parameters and can be excluded from the likelihood. Merge in expected mortality rate at time of death, h (t i ). Newton-Raphson algorithm implemented using Stata ml command (method lf). Incorporating delayed entry allows period analysis models to be fitted[8]. This is a method used to obtain up-to-date estimates of (relative) survival. Application in the cure models allows up-to-date estimates of cure to be obtained. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 11/27

26 strsmix and strsnmix commands strsmix [ varlist ] [ if ] [ in ], distribution(distribution) link(link function) bhazard(varname) [ k1(varlist) k2(varlist) k3(varlist) k4(varlist) pmix(varlist) noconstant noconsk1 noconsk2 noconsk3 noconsk4 noconspmix init(matrix name) skip inititer(#) stopconstraint valconstraint(#) eform ] strsnmix [ varlist ] [ if ] [ in ], distribution(distribution) link(link function) bhazard(varname) [ k1(varlist) k2(varlist) k3(varlist) k4(varlist) pmix(varlist) split(#) earlyk1(varlist) earlyk2(varlist) noconstant noconsk1 noconsk2 noconsk3 noconsk4 noconspmix earlynoconsk1 earlynoconsk2 init(matrix name) skip inititer(#) stopconstraint valconstraint(#) eform ] Stata net from install strsnmix Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 12/27

27 Some options for strsnmix and strsmix distribution(distribution) specifies the parametric distribution. Arguments for both strsmix and strsnmix are weibull, lognomal and gamma, weibexp and weibweib. link(link function) specifies the link function for the cure fraction. Options are identity, logistic and loglog. Note that loglog is ln( ln(π)). bhazard(varname) gives the variable name for the baseline hazard at death/censoring. This option is compulsory, but standard cure models can be estimated by making varname a column of zeros. k1-k4(varlist) gives any covariates for the auxillary parameter. E.g. for the Weibull distribution k1 refers to ln(λ) and k2 refers to ln(γ). Commands submitted to The Stata Journal[5]. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 13/27

28 Cancer of the Colon in Finland Data from the Finnish Cancer Registry. 27,754 men and women diagnosed with follow-up to Covariates age group and year of diagnosis. Exclude those aged 80 years and over. Use a mixture cure model with Weibull distribution for the uncured. Year of diagnosis modelled using restricted cubic splines for cure fraction and both Weibull parameters. Stata Code strsmix rcs1-rcs4 agegrp2 agegp3 agegrp4 age2rcs1 age3rcs1 age4rcs1, /// dist(weibull) link(identity) bhazard(brate) /// k1(rcs1-rcs4 agegrp2 agegrp3 agegrp4 age2rcs1 age3rcs1 age4rcs1) /// k2(rcs1-rcs4 agegrp2 agegrp3 agegrp4 age2rcs1 age3rcs1 age4rcs1) predict cure, cure ci predict rs, survival ci predict rsu, survival uncured ci predict exhaz, hazard ci predict median, centile ci Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 14/27

29 Time Trends for Cancer of the Colon Age < Cure Fraction and Median Survival of Uncured Age Group: <50 Cure Fraction Median Survival Cure Fraction Median Survival Year of Diagnosis 0.0 Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 15/27

30 Time Trends for Cancer of the Colon Age < Cure Fraction and Median Survival of Uncured Age Group: <50 Cure Fraction Median Survival Cure Fraction Median Survival Year of Diagnosis 0.0 Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 15/27

31 Time Trends for Cancer of the Colon <50 years years years years Cure Fraction Cure Fraction Year of Diagnosis Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 16/27

32 Time Trends for Cancer of the Colon <50 years years years years Median Survival of Uncured Median Survival of Uncured Year of Diagnosis Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 16/27

33 Quantifying Differences 0.2 Difference in Cure Fraction (Age Group <50 Age Group 70 79) Difference in Cure Fraction Years from Diagnosis Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 17/27

34 Period Analysis Long-term estimates of survival may be out-of-date. Period Analysis estimates (relative) survival by only incorporating survival experience in a recent time window[1]. Period Analysis generally estimated in lifetables, but simple to incorporate in to modelling environment[8]. In survival models period analysis can be incorporated using delayed entry techniques. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 18/27

35 Period Analysis Subject 1 Diagnosis Death or Censoring Start and Stop at Risk Times Standard Period (0, 2) Subject 2 (0, 4) Subject 3 Subject Year (0, 6) (0, 3) Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 19/27

36 Period Analysis Subject 1 Diagnosis Death or Censoring Start and Stop at Risk Times Standard Period (0, 2) Subject 2 (0, 4) Subject 3 Subject 4 Period of Interest Year (0, 6) (0, 3) Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 19/27

37 Period Analysis Subject 1 Diagnosis Death or Censoring Start and Stop at Risk Times Standard Period (0, 2) (0, 2) Subject 2 (0, 4) (2, 4) Subject 3 Subject 4 Period of Interest Year (0, 6) (0, 3) (5, 6) (, ) Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 19/27

38 Period Analysis: Cancer of the Colon.6 Cancer of the Colon: Cure Fraction.5 Cure Fraction Standard Lag 10 years Period Analysis Model 1 Model 2 Model Year Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 20/27

39 Period Analysis: Cancer of the Colon.6 Cancer of the Colon: Cure Fraction.5 Cure Fraction Standard Lag 10 years Period Analysis Model 1 Model 2 Model Year Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 20/27

40 Period Analysis: Cancer of the Colon.6 Cancer of the Colon: Cure Fraction.5 Cure Fraction Standard Lag 10 years Period Analysis Model 1 Model 2 Model Year Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 20/27

41 Period Analysis: Cancer of the Colon.6 Cancer of the Colon: Cure Fraction.5 Cure Fraction Standard Lag 10 years Period Analysis Model 1 Model 2 Model Year Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 20/27

42 Period Analysis: Cancer of the Colon.6 Cancer of the Colon: Cure Fraction.5 Cure Fraction Standard Lag 10 years Period Analysis Model 1 Model 2 Model Year Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 20/27

43 Period Analysis: Cancer of the Colon.6 Cancer of the Colon: Cure Fraction.5 Cure Fraction Standard Lag 10 years Period Analysis Model 1 Model 2 Model Year Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 20/27

44 More Flexible Models In some situations the Weibull distribution is not flexible enough and results in a poor fit. Usually when very high excess mortality rate in first few weeks after diagnosis. Other, more flexible, distributions can be considered LogNormal and Generalized Gamma are implemented LogNormal fits poorly due to Long tail Some Convergence problems with Generalized Gamma Two Extensions Split-time models. These split the time scale into two. Within the first time interval (up to time k) use simple parametic model for the relative survival and then fit a cure fraction model condition on survival to time k. Use a Finite Mixture of Distributions. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 21/27

45 More Flexible Models In some situations the Weibull distribution is not flexible enough and results in a poor fit. Usually when very high excess mortality rate in first few weeks after diagnosis. Other, more flexible, distributions can be considered LogNormal and Generalized Gamma are implemented LogNormal fits poorly due to Long tail Some Convergence problems with Generalized Gamma Two Extensions Split-time models. These split the time scale into two. Within the first time interval (up to time k) use simple parametic model for the relative survival and then fit a cure fraction model condition on survival to time k. Use a Finite Mixture of Distributions. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 21/27

46 More Flexible Models In some situations the Weibull distribution is not flexible enough and results in a poor fit. Usually when very high excess mortality rate in first few weeks after diagnosis. Other, more flexible, distributions can be considered LogNormal and Generalized Gamma are implemented LogNormal fits poorly due to Long tail Some Convergence problems with Generalized Gamma Two Extensions Split-time models. These split the time scale into two. Within the first time interval (up to time k) use simple parametic model for the relative survival and then fit a cure fraction model condition on survival to time k. Use a Finite Mixture of Distributions. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 21/27

47 Mixture of Distributions Non-Mixture Model h(t) = h (t) ln(π)(pf z1 (t) + (1 p)f z2 (t)) This allows a much more flexible shape for the excess hazard and relative survival function[11]. Mixture of two Weibull distributions generally works well. Can also think of two groups of individuals, those who die after a short time and those who die after a longer time. Mixture Model S(t) = S (t)(π + (1 π)(ps u1 (t) + (1 p)s u2 (t))) For mixture models on relative survival scale. Mixture of two Weibull distributions generally works well. Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 22/27

48 Cancer of the Colon: Weibull and Mixture of Weibulls Age Group Ederer II Weibull Mixture of Weibulls Relative Survival Years from Diagnosis Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 23/27

49 Cancer of the Colon: Weibull and Mixture of Weibulls Age Group Ederer II Weibull Mixture of Weibulls Relative Survival Years from Diagnosis Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 23/27

50 Cancer of the Colon: Weibull and Mixture of Weibulls Age Group Ederer II Weibull Mixture of Weibulls Relative Survival Years from Diagnosis Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 23/27

51 Cancer of the Colon: Weibull and Mixture of Weibulls Age Group 80+ Ederer II Weibull Mixture of Weibulls Relative Survival Years from Diagnosis Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 24/27

52 Cancer of the Colon: Weibull and Mixture of Weibulls Age Group 80+ Ederer II Weibull Mixture of Weibulls Relative Survival Years from Diagnosis Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 24/27

53 Cancer of the Colon: Weibull and Mixture of Weibulls Age Group 80+ Ederer II Weibull Mixture of Weibulls Relative Survival Years from Diagnosis Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 24/27

54 Cancer of the Colon: Excess Hazard Rate 1.5 Age Group 80+ Survival of Uncured (Weibull) Survival of Uncured (Mixture of Weibulls) Excess hazard rate Years from Diagnosis Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 25/27

55 Cancer of the Colon: Excess Hazard Rate Age Group Excess Hazard Rate (Mixture of Weibulls) Mixture Component 1 (37%) Mixture Component 2 (63%) Excess hazard rate Years from Diagnosis Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 25/27

56 Summary In population based cancer studies cure is often observed. Relative survival models that explicitly allow for cure are useful for monitoring trends and differences in (relative) survival. strsnmix and strsmix fit a wide range of models. Incorporation of delayed entry models allows up-to-date estimates of cure to be obtained. Still needs to be a degree of caution When cure is not a reasonable assumption. Follow-up not long enough. Simple models may not fit the data well, but alternatives are available. When the cure fraction is high (over 75-80%). Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 26/27

57 [1] H. Brenner and O. Gefeller. Deriving more up-to-date estimates of long-term patient survival. Journal of Clinical Epidemiology, 50(2): , [2] R. De Angelis, R. Capocaccia, T. Hakulinen, B. Soderman, and A. Verdecchia. Mixture models for cancer survival analysis: application to population-based data with covariates. Statistics in Medicine, 18(4):441 54, [3] P.W. Dickman, A. Sloggett, M. Hills, and T. Hakulinen. Regression models for relative survival. Statistics in Medicine, 23:41 64, [4] R. Giorgi, M. Abrahamowicz, C. Quantin, P. Bolard, J. Esteve, J. Gouvernet, and J. Faivre. A relative survival regression model using b-spline functions to model non-proportional hazards. Statistics in Medicine, 22(17): , [5] P.C. Lambert. Estimating and modelling the cure fraction in population-based cancer survival analysis. Submitted to The Stata Journal, [6] P.C. Lambert, L. K. Smith, D. R. Jones, and J.L. Botha. Additive and multiplicative covariate regression models for relative survival incorporating fractional polynomials for time-dependent effects. Statistics in Medicine, 24: , [7] P.C. Lambert, J.R. Thompson, C. L. Weston, and P. W. Dickman. Estimating and modelling the cure fraction in population-based cancer survival analysis. Submitted to Biostatistics, [8] L. K. Smith, P. C. Lambert, J. L. Botha, and D. R. Jones. Providing more up-to-date estimates of patient survival: a comparison of standard survival analysis with period analysis using life-table methods and proportional hazards models. Journal of Clinical Epidemiology, 57(1):14 20, [9] R. Sposto. Cure model analysis in cancer: an application to data from the children s cancer group. Statistics in Medicine, 21(2): , [10] A. Verdecchia, R. De Angelis, R. Capocaccia, M. Sant, A. Micheli, G. Gatta, and F. Berrino. The cure for colon cancer: results from the eurocare study. International Journal of Cancer, 77(3):322 9, [11] A.Y. Yakovlev, A.D. Tsodikov, K. Boucher, and R. Kerber. The shape of the hazard function in breast carcinoma: Curability of the disease revisited. Cancer, 85: , Paul C Lambert Cure Models and Relative Survival 12th UK Stata Users Group 27/27